Projection pair: Difference between revisions
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A '''projection pair''' is a pair of two rational intervals which can be employed by the [[Scala]] "project" command to reduce a JI scale to a scale in a [[JI subgroup]] of the group generated by the scale, in such a way that tempered versions of each are equivalent. This is particularly useful for analyzing [[planar temperament]]s, as the projection can then be viewed in lattice form by Scala's "lattice" or "lattice and player" command. | A '''projection pair''' is a pair of two rational intervals which can be employed by the [[Scala]] "project" command to reduce a JI scale to a scale in a [[JI subgroup]] of the group generated by the scale, in such a way that tempered versions of each are equivalent. This is particularly useful for analyzing [[planar temperament]]s, as the projection can then be viewed in lattice form by Scala's "lattice" or "lattice and player" command. | ||
An example of a projection pair is "7 225/32", which when applied by Scala's "project" to a 7-limit scale produces a 5-limit scale, which when tempered by marvel (225/224) temperament gives exactly the same result as the original scale does when also tempered. More than one such pair may be required to reduce to the desired subgroup; for instance "7 225/32 11 4096/375" reduces an 11-limit JI scale to a 5-limit JI scale equivalent under (undecimal) marvel. This can happen even when only one comma is involved (codimension one temperaments). For instance, to project a 7-limit scale in the hemimean (3136/3125) reduction to the 2.5.7 subgroup requires "5 3136/625 7 68841472/9765625". | An example of a projection pair is "7 225/32", which when applied by Scala's "project" to a 7-limit scale produces a 5-limit scale, which when tempered by marvel (225/224) temperament gives exactly the same result as the original scale does when also tempered. More than one such pair may be required to reduce to the desired subgroup; for instance "7 225/32, 11 4096/375" reduces an 11-limit JI scale to a 5-limit JI scale equivalent under (undecimal) marvel. This can happen even when only one comma is involved (codimension one temperaments). For instance, to project a 7-limit scale in the hemimean (3136/3125) reduction to the 2.5.7 subgroup requires "5 3136/625, 7 68841472/9765625". | ||
Many projection pairs are given on the pages for various planar temperaments. When no subgroup is indicated, the default 2.3.5 5-limit subgroup is presumed. These lists of pairs can be copied and pasted into Scala and applied to any suitable JI scale. | Many projection pairs are given on the pages for various planar temperaments. When no subgroup is indicated, the default 2.3.5 5-limit subgroup is presumed. These lists of pairs can be copied and pasted into Scala and applied to any suitable JI scale. | ||
== List of 5-limit | == List of projection pairs == | ||
* 16875/16384: 3 50625/16384 5 16384/3375 to 2.15 | === 5-limit === | ||
* 250/243: 3 729/250 5 59049/12500 to 2.9/5 | * 16875/16384: 3 50625/16384, 5 16384/3375 to 2.15 | ||
* 250/243: 3 729/250, 5 59049/12500 to 2.9/5 | |||
* 3125/3072: 3 3125/1024 | * 3125/3072: 3 3125/1024 | ||
* 20000/19683: 3 20000/6561 5 2000000000/387420489 to 2.9/5 | * 20000/19683: 3 20000/6561, 5 2000000000/387420489 to 2.9/5 | ||
* 81/80: 5 81/16 | * 81/80: 5 81/16 | ||
* 393216/390625: 3 390625/131072 | * 393216/390625: 3 390625/131072 | ||
* 15625/15552: 3 46656/15625 5 15552/3125 to 2.5/3 | * 15625/15552: 3 46656/15625, 5 15552/3125 to 2.5/3 | ||
* 32805/32768: 5 32768/6561 | * 32805/32768: 5 32768/6561 | ||
== | === 7-limit === | ||
* 1029/1000: 3 1000/343 to 2.5.7 | * 1029/1000: 3 1000/343 to 2.5.7 | ||
* 36/35: 7 36/5 | * 36/35: 7 36/5 | ||
* 525/512: 7 512/75 | * 525/512: 7 512/75 | ||
* 49/48: 3 49/16 to 2.5.7 | * 49/48: 3 49/16 to 2.5.7 | ||
* 686/675: 5 3375/686 7 675/98 to 2.3.7/5 | * 686/675: 5 3375/686, 7 675/98 to 2.3.7/5 | ||
* 64/63: 7 64/9 | * 64/63: 7 64/9 | ||
* 854296875/843308032: 5 843308032/170859375 7 5903156224/854296875 to 2.3.7/5 | * 854296875/843308032: 5 843308032/170859375, 7 5903156224/854296875 to 2.3.7/5 | ||
* 64827/64000: 5 320000/64827 7 64000/9261 to 2.3.7/5 | * 64827/64000: 5 320000/64827, 7 64000/9261 to 2.3.7/5 | ||
* 875/864: 7 864/125 | * 875/864: 7 864/125 | ||
* 3125/3087: 5 15625/3087 7 9765625/1361367 to 2.3.25/7 | * 3125/3087: 5 15625/3087, 7 9765625/1361367 to 2.3.25/7 | ||
* 2430/2401: 5 2401/486 to 2.3.7 | * 2430/2401: 5 2401/486 to 2.3.7 | ||
* 50421/50000: 3 50000/16807 to 2.5.7 | * 50421/50000: 3 50000/16807 to 2.5.7 | ||
* 245/243: 5 243/49 to 2.3.7 | * 245/243: 5 243/49 to 2.3.7 | ||
* 126/125: 7 125/18 | * 126/125: 7 125/18 | ||
* 4000/3969: 5 3969/800 7 27783/4000 to 2.3.7/5 | * 4000/3969: 5 3969/800, 7 27783/4000 to 2.3.7/5 | ||
* 1728/1715: 5 1728/343 to 2.3.7 | * 1728/1715: 5 1728/343 to 2.3.7 | ||
* 1029/1024: 3 1024/343 to 2.5.7 | * 1029/1024: 3 1024/343 to 2.5.7 | ||
* 225/224: 7 225/32 | * 225/224: 7 225/32 | ||
* 19683/19600: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7 | * 19683/19600: 3 19600/6561, 7 1033052339200000000/150094635296999121 to 2.5.81/7 | ||
* 16875/16807: 5 84375/16807 7 16875/2401 to 2.3.7/5 | * 16875/16807: 5 84375/16807, 7 16875/2401 to 2.3.7/5 | ||
* 10976/10935: 5 10976/2187 to 2.3.7 | * 10976/10935: 5 10976/2187 to 2.3.7 | ||
* 3136/3125: 5 3136/625 7 68841472/9765625 to 2.3.25/7 | * 3136/3125: 5 3136/625, 7 68841472/9765625 to 2.3.25/7 | ||
* 5120/5103: 7 5120/729 | * 5120/5103: 7 5120/729 | ||
* 6144/6125: 3 6125/2048 to 2.5.7 | * 6144/6125: 3 6125/2048 to 2.5.7 | ||
| Line 43: | Line 44: | ||
* 201768035/201326592: 5 201326592/40353607 to 2.3.7 | * 201768035/201326592: 5 201326592/40353607 to 2.3.7 | ||
* 65625/65536: 7 65536/9375 | * 65625/65536: 7 65536/9375 | ||
* 703125/702464: 5 702464/140625 7 3454189699072/494384765625 to 2.3.25/7 | * 703125/702464: 5 702464/140625, 7 3454189699072/494384765625 to 2.3.25/7 | ||
* 420175/419904: 5 882735153125/176319369216 7 419904/60025 to 2.3.245 | * 420175/419904: 5 882735153125/176319369216, 7 419904/60025 to 2.3.245 | ||
* 2401/2400: 3 2401/800 to 2.5.7 | * 2401/2400: 3 2401/800 to 2.5.7 | ||
* 4375/4374: 7 4374/625 | * 4375/4374: 7 4374/625 | ||
=== 11-limit === | |||
* 99/98: 11 98/9 | |||
* 100/99: 11 100/9 | |||
* 121/120: 5 121/24 to 2.3.11 | |||
* 176/175: 11 175/16 | |||
* 243/242: 3 242/81, 11 644204/59049 to 2.11/9 | |||
* 385/384: 11 384/35 | |||
* 441/440: 11 441/40 | |||
* 540/539: 11 540/49 | |||
* 3025/3024: 7 3025/432 to 2.3.5.11 | |||
[[Category:Lists of intervals]] | [[Category:Lists of intervals]] | ||
[[Category:Just intonation]] | [[Category:Just intonation]] | ||
[[Category:Method]] | [[Category:Method]] | ||